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Please Help ! Number Theory

by pullagurla » Tue Jul 13, 2010 10:08 am
1. A number when divided by 221 leaves a remainder 64 . what is the remainder if the same number is divided by 13
(a)0 (b)11 (c)7 (d)1 (e)12

2. Two different numbers when divided by same divisor leaves remainder 3 and 4 respectively and when their sum is divided by the same divisor, remainder is 1. What is the divisor ?
a)4 b)5 c)6 d)7 e)8

3. How many times should the keys of a type writer be pressed in order to type first 400 counting numbers ?
a) 1082 b) 1092 c)1192 d)1200 e)12082

4) How many of the Odd Integers between 10 and 20 can be expressed as the difference between the squares of two consecutive integers
a)5 b)4 c)3 d)1 e)0

5) The traffic lights at three different road crossings change after every 48sec , 72 sec, and 108 sec respectively . if they all change simultaneously at 8:20:00 hours, they will again change simultaneously at ?
a) 8:27:12 hrs b)8:27:48 hrs c)8:27:36 hrs d)8:27:24 hrs e)none[/list]

GIVE EXPLANATIONS PLEASE :)
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by outreach » Tue Jul 13, 2010 10:18 am
1

answer should be E


2
there is a theorem which say
when a number 'a' is divided by a divisor d and remainder is r1
when a number 'b' is divided by a divisor d and remainder is r2

when sum of a+b is divided by a divisor d and remainder is r1+r2.
if r1+r2>d then remainder is r1+r2-d

hence answer should be 6
pullagurla wrote:1. A number when divided by 221 leaves a remainder 64 . what is the remainder if the same number is divided by 13
(a)0 (b)11 (c)7 (d)1 (e)12

2. Two different numbers when divided by same divisor leaves remainder 3 and 4 respectively and when their sum is divided by the same divisor, remainder is 1. What is the divisor ?
a)4 b)5 c)6 d)7 e)8

GIVE EXPLANATIONS PLEASE :)
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by GMATGuruNY » Tue Jul 13, 2010 10:23 am
pullagurla wrote:1. A number when divided by 221 leaves a remainder 64 . what is the remainder if the same number is divided by 13
(a)0 (b)11 (c)7 (d)1 (e)12
A number when divided by 221 leaves a remainder of 64 means that the number is 64 more than a multiple of 221.

The smallest positive multiple of 221 is 221. (Every number is a multiple of itself.)

So let's make the number 221 + 64 = 285. This works because 285/221 = 1 R64.

285/13 = 21 R12.

The correct answer is E.
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by GMATGuruNY » Tue Jul 13, 2010 10:30 am
pullagurla wrote:
2. Two different numbers when divided by same divisor leaves remainder 3 and 4 respectively and when their sum is divided by the same divisor, remainder is 1. What is the divisor ?
a)4 b)5 c)6 d)7 e)8

GIVE EXPLANATIONS PLEASE :)
Let's try the answers, which represent the possible divisor. We should start with C, the answer choice in the middle. C says that the divisor is 6.

If the divisor is 6 and the remainder is 3, one of the numbers must be 3 more than a multiple of 6. Let's use 6 + 3 = 9 for this number.

If the divisor is 6 and the remainder is 4, the other number must be 4 more than a multiple of 6. Let's use 6 + 4 = 10 for the other number.

The sum of the 2 numbers is 9 + 10 = 19. When we divide by 6, is the remainder 1? Yes, because 19/6 = 3 R1. So answer choice C satisfies all the conditions in the problem.

The correct answer is C.
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by outreach » Tue Jul 13, 2010 10:36 am
4
odd no=11 13 15 17 19

odd = consecutive no which can be squared
11 =6^2-5^2
13= 7,6
15=7,8
17=9,8
19=10,9


5
lcm of 48=2*2*2*2*3
72=2*2*2*3*3
108=2*2*3*3*3



2*2*2*2*3*3*3
=16*27=432 sec
=7.12min

A
pullagurla wrote: 4) How many of the Odd Integers between 10 and 20 can be expressed as the difference between the squares of two consecutive integers
a)5 b)4 c)3 d)1 e)0

5) The traffic lights at three different road crossings change after every 48sec , 72 sec, and 108 sec respectively . if they all change simultaneously at 8:20:00 hours, they will again change simultaneously at ?
a) 8:27:12 hrs b)8:27:48 hrs c)8:27:36 hrs d)8:27:24 hrs e)none[/list]

GIVE EXPLANATIONS PLEASE :)
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by barcebal » Tue Jul 13, 2010 10:44 am
4) How many of the Odd Integers between 10 and 20 can be expressed as the difference between the squares of two consecutive integers
a)5 b)4 c)3 d)1 e)0
I know the explanation has been posted, but this has helped me:

The difference between two consecutive squares is (1) always an odd number and (2) will always lie in the set (1,3,5,7,....)

For example
|
|
|
V
1^2=1
Difference---->3
2^2=4
Difference---->5
3^2=9
Difference---->7
4^2=16
Difference---->11
5^2=25
Difference---->13
6^2=36
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by kvcpk » Tue Jul 13, 2010 10:47 am
Hi Pullagurla,

Request you to post only one question per post. Post 5 different posts for 5 questions. It will be easy for the people answering them.

Coming to your queries:

1. A number when divided by 221 leaves a remainder 64 . what is the remainder if the same number is divided by 13
(a)0 (b)11 (c)7 (d)1 (e)12

A number when divided by 221 leaves a remainder 64 means that the number is divisible by 221 +64 =285
285 when divided by 13 will give reminder12. pick E.

2. Two different numbers when divided by same divisor leaves remainder 3 and 4 respectively and when their sum is divided by the same divisor, remainder is 1. What is the divisor ?
a)4 b)5 c)6 d)7 e)8
I would prefer going by options.
take 4: divisor cannot be 4, because reminder cannot be 4 when divisor is 4.
take 5: let x=8, y=9 [they satisfy initial remainders]
8+9 = 17 when div by 5 rem is 2 .. hence NO
take 6: let x=9, y=10
9+10=19 when div by 6, rem is 1
pick C.

3. How many times should the keys of a type writer be pressed in order to type first 400 counting numbers ?
a) 1082 b) 1092 c)1192 d)1200 e)12082
number of 1 digit nos + 2*(number of 2 digit nos)+3*(number of 3 digit nos)
one digit numbers = 9
two digit numbers = 90
three digit numbers = 301
9+180+903 = 1092 .. pick B.

4) How many of the Odd Integers between 10 and 20 can be expressed as the difference between the squares of two consecutive integers
a)5 b)4 c)3 d)1 e)0
odd integers = 11,13,15,17,19
(x+1)^2-x^2
2x+1
need to see how many of the above list will satisfy 2x+1
put x=5 2x+1=11
x=6 -> 13
x=7 -> 15
x=8 -> 17
x=9 -> 19

means 5. Pick A.

5) The traffic lights at three different road crossings change after every 48sec , 72 sec, and 108 sec respectively . if they all change simultaneously at 8:20:00 hours, they will again change simultaneously at ?
a) 8:27:12 hrs b)8:27:48 hrs c)8:27:36 hrs d)8:27:24 hrs e)none

LCM of 48,72,108 is 432.
So for every 432 seconds, crossings change together.
432 = 7mins 12 seconds
pick A.
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by GMATGuruNY » Tue Jul 13, 2010 10:48 am
pullagurla wrote: 3. How many times should the keys of a type writer be pressed in order to type first 400 counting numbers ?
a) 1082 b) 1092 c)1192 d)1200 e)12082

GIVE EXPLANATIONS PLEASE :)
To count the total number of consecutive integers in a given list, use the following formula:

total number of consecutive integers = biggest - smallest + 1

Let's count the number of digits in all the integers 1 - 400.

Single digit integers 1-9:

9 - 1 + 1 = 9 integers. Each has 1 digit. So 9 digits total.

Two-digit integers 10-99:

99 - 10 + 1= 90 integers. Each has 2 digits. 2 * 90 = 180 digits total.

Three-digit integers 100-400:

400 - 100 + 1 = 301 integers. Each has 3 digits. 3 * 301 = 903 digits total.

So total digits = 9 + 180 + 903 = 1092.

The correct answer is B.
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by GMATGuruNY » Tue Jul 13, 2010 11:01 am
pullagurla wrote: 4) How many of the Odd Integers between 10 and 20 can be expressed as the difference between the squares of two consecutive integers
a)5 b)4 c)3 d)1 e)0

GIVE EXPLANATIONS PLEASE :)
Let's make a list of perfect squares (perfect square means the square of an integer):

1^2 = 1
2^2 = 4
3^3 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81
10^2=100
11^2 = 121

We can stop there because 121-100=21, and we want the difference to be between 10 and 20.
25-16=9, which is too small, so all the viable pairs will be between 5^2=36 and 10^2=100.

So only 5 pairs will work:

6^2 = 36, 5^2 = 25, 36-25=11.
7^2 = 49, 6^2 = 36, 49-36=13.
8^2=64, 7^2=49, 64-49=15.
9^2=81, 8^2=64, 81-64=17.
10^2=100, 9^2=81, 100-81=19.

The correct answer is A.
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by GMATGuruNY » Tue Jul 13, 2010 11:22 am
pullagurla wrote:
5) The traffic lights at three different road crossings change after every 48sec , 72 sec, and 108 sec respectively . if they all change simultaneously at 8:20:00 hours, they will again change simultaneously at ?
a) 8:27:12 hrs b)8:27:48 hrs c)8:27:36 hrs d)8:27:24 hrs e)none[/list]

GIVE EXPLANATIONS PLEASE :)
We need the smallest number that is a multiple of 48, 72 and 108.

Let's try increasing multiples of the largest number (108) until we find one that is also divisible by 48 and 72.

108*1 = 108. 108/48 = 2.25 Doesn't work.
108*2 = 216. 216/48 = 4.5 Doesn't work.
108*3 = 324. 324/48 = 6.75 Doesn't work.
108*4 = 432. 432/48 = 9. This works! And 432/72 = 6, so 432 is the smallest number that is a multiple of 48, 72, and 108.

432/60 = 7 minutes, 12 seconds.

8:20:00 + 0:7:12 = 8:27:12.

The correct answer is A.
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